scholarly journals Planar Typical Bézier Curves Made Simple

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3017
Author(s):  
Javier Sánchez-Reyes

Recently, He et al. derived several remarkable properties of the so-called typical Bézier curves, a subset of constrained Bézier curves introduced by Mineur et al. In particular, He et al. proved that such curves display at most one curvature extremum, give an explicit formula of the parameter at the extremum, and show that subdividing a curve at this point furnishes two new typical curves. We recall that typical curves amount to segments of a special family of sinusoidal spirals, curves already studied by Maclaurin in the early 18th century and whose properties are well-known. These sinusoidal spirals display only one curvature extremum (i.e., vertex), whose parameter is simply that corresponding to the axis of symmetry. Subdividing a segment at an arbitrary point, not necessarily the vertex, always yields two segments of the same spiral, hence two typical curves.

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2148
Author(s):  
Chuan He ◽  
Gang Zhao ◽  
Aizeng Wang ◽  
Shaolin Li ◽  
Zhanchuan Cai

This paper focuses on planar typical Bézier curves with a single curvature extremum, which is a supplement of typical curves with monotonic curvature by Y. Mineur et al. We have proven that the typical curve has at most one curvature extremum and given a fast calculation formula of the parameter at the curvature extremum. This will allow designers to execute a subdivision at the curvature extremum to obtain two pieces of typical curves with monotonic curvature. In addition, we put forward a sufficient condition for typical curve solutions under arbitrary degrees for the G1 interpolation problem. Some numerical experiments are provided to demonstrate the effectiveness and efficiency of our approach.


2020 ◽  
Vol 75 (1) ◽  
pp. 323-378
Author(s):  
David Allen ◽  
Briony A. Lalor ◽  
Ginny Pringle

This report describes excavations at Basing Grange, Basing House, Hampshire, between 1999 and 2006. It embraces the 'Time Team' investigations in Grange Field, adjacent to the Great Barn, which were superseded and amplified by the work of the Basingstoke Archaeological & Historical Society, supervised by David Allen. This revealed the foundations of a 'hunting lodge' or mansion built in the 1670s and demolished, and effectively 'lost', in the mid-18th century. Beneath this residence were the remains of agricultural buildings, earlier than and contemporary with the nearby Great Barn, which were destroyed during the English Civil War. The report contains a detailed appraisal of the pottery, glass and clay tobacco pipes from the site and draws attention to the remarkable window leads that provide a clue to the mansion's date of construction. It also explores a probable link with what was taking place on the Basing House site in the late 17th and early 18th century.


2021 ◽  
Author(s):  
Satyanarayana G. Manyam ◽  
David Casbeer ◽  
Isaac E. Weintraub ◽  
Dzung M. Tran ◽  
Justin M. Bradley ◽  
...  

2021 ◽  
Vol Accepted ◽  
Author(s):  
Bayram Şahin ◽  
Aslı Ayar

2018 ◽  
Vol 2018/2 ◽  
pp. 31-53

DESIGNATION OF JUDICIAL DOCUMENTS IN THE THIRD STATUTE OF LITHUANIA AND THE ATTRIBUTES OF THEIR EVOLUTION ADAM STANKEVIČ The author of the article analyses the designation of documents drawn up and issued by the court, their conception, field of application, and place in the court procedure as presented in the Third Statute of Lithuania (TSL). In addition, an attempt is made to exhibit the changes that such documents and their designations underwent in later centuries (until the end of the 18th c.) by means of the example of the Lithuanian Tribunal. The research revealed that documents which in the sources from different periods were referred to by the same name meant different things or were simultaneously attributed several meanings. In the 17th-18th century, only part of the terms featured in the Third Statute of Lithuania were used in the judicial practice of the Lithuanian Tribunal, and with time some of them were replaced with other terms. Several terms denoting summonses (pozew, mandat, zakaz) can be identified in the TSL, and all of them were in use until the very end of the 18th century. However, a single term – dekret / decretum – was used to designate the judgement (actually, for some time there was a differentiation between the court judgement and its procedural summary, but later the generalized term for the judgement prevailed). A number of documents in the TSL are referred to as the “open letter”, however, later some of them acquired specialised names (e.g. the power of attorney). With time, there were certain changes in the context in which some of the terms were used (e.g. the term “cedule” which in the 18th century was already consistently used exceptionally in a particular situation, namely when a litigant refused to obey the order of the court and informed in writing a judicial officer of such refusal) or the terms themselves underwent certain changes (in the 18th century the term membran was substituted with the term blankiet). Part of the judicial documents mentioned in the TSL disappeared in the long run or there was a certain decrease in their significance (this is true of the reminder and adjournment documents as well as glejt (protection letter)). The examples above suggest that the Lithuanian Tribunal would sometimes issue reminders and guarantee documents, though legal acts did not explicitly provide for that. The TSL offered a number of terms hardly related with the investigation of a case, therefore in the early 18th century, with the improvement of judicial procedures, they underwent rapid changes. The procedure of the implementation of a court ruling, which underwent significant changes, is accountable for the introduction of new terms, for example, with time several terms pertaining to the notification of the litigants were used simultaneously (obwieszczenie, innotescencyja, list tradycyjny). Most probably due to the unification processes observed in the Polish-Lithuanian Commonwealth in the 18th century, a number of Latin origin terms were introduced in the judicial practice of the GDL, e.g. cytacyja, decyzyja, innotestencyja, plenipotencyja, obdukcyja, wizyja, inkwizycyja, weryfikacyja, kalkulacyja, tradycyja (all of them had been used in Poland but were not featured in the TSL).


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